Printers and certain other output devices having limited bit depth can only produce a limited number or level of primaries per location. Consider, for example, a printer that prints with cyan (C), magenta (M), yellow (Y), black (k) inks. If the printer has limited bit depth of 1, at a certain location it can deposit only 21 (i.e., 2) levels of primaries. Either the printer deposits a full level (a dot of ink) or a zero level (no ink) at each location. To create a non-white color patch on a white print medium, dots of one or more of these inks are deposited within the patch. The human visual system does not perceive the individual ink dots in the color patch, but rather perceives the spatial average color of the ink dots. Consequently, the color patch is perceived as having continuous tones when viewed through the human visual system.
A limited bit-depth output device can be modeled by a transfer function, which specifies the color produced from a given combination of primaries. For example, the transfer function of a four colorant printer might specify the color that results from printing with 20% magenta and 20% cyan. The transfer function provides a many-to-one mapping of primary space to desired color space, since different combinations of primaries can produce the same desired color.
The inverse transfer function specifies the relative amounts of primaries that produce a desired color. For example, the inverse transfer function of a four-colorant printer can specify the relative amounts of cyan and magenta to produce a desired shade of blue. Typically found during product development, the inverse transfer function of the four-colorant printer is used to convert continuous tone images into halftone images that can be printed by the four-colorant printer.
Finding the inverse transfer function is well established for four-color printers. The inverse transfer functions for four-color printers are derived empirically. A large knowledge base already exists for four color printers.
However, the large knowledge base does not extend to printers having more than four primaries, nor does it extend to limited bit-depth output devices other than printers. For limited bit depth output devices that have more than four primaries, additional information is needed. The additional information can be obtained by trial and error, and the inverse transfer functions can be determined by ad-hoc methods and empirical fixes. However, the trial error can be expensive and time consuming. The expense and time increases exponentially with an increase in the number of primaries.
Moreover, finding the inverse transfer function of a limited bit depth output device having more than four primary colorants is an ill-posed problem. The problem is ill-posed because the desired colors are usually expressed in three-dimensional space.
It would be desirable to mathematically determine the inverse transfer function of an output device having more than four primaries. High fidelity printers, which use more than four distinct primary colorants to obtain better color reproduction, are becoming more popular. Some high fidelity printers use CMYK, O(orange) and G(green) inks, others use CMYKO and V (violet) and still others use CMYK, R (red), G and B(blue) inks. Currently, inverse transfer functions for these high fidelity printers are determined by ad-hoc methods and empirical fixes. One advantage of the mathematical description is that it would reduce the amount of empirical trial and error and, thereby, increase the speed and reduce the cost of product development.